Problem: Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$.
Explanation: The first cube root becomes $\sqrt[3]{9}$. $\sqrt[3]{8}=2$, so the second cube root becomes $\sqrt[3]{3}$. Multiplying these gives $\sqrt[3]{27} = \boxed{3}$.